A Note on Preprojective Partitions Over Hereditary Artin Algebras

Abstract

If $\Lambda$ is an artin algebra there is a partition of $\operatorname {ind} \Lambda$, the category of indecomposable finitely generated $\Lambda$-modules, $\operatorname {ind} \Lambda = { \cup _{i \geqslant 0}}{\underline {\underline {P}}_i}$, called the preprojective partition. We show that $\underline {\underline {P}}_i$ can be easily constructed for hereditary artin algebras, if $\underline {\underline {P}}_{i - 1}$ is known: $A$ is in $\underline {\underline {P}}_i$ if and only if $A$ is not in $\underline {\underline {P}}_{i - 1}$ and there is an irreducible map $B \to A$, where $B$ is in $\underline {\underline {P}}_{i - 1}$.